Solid State Theory Physics 545 Symmetry and Introduction to Group Theory

StillddifdtltftSymmetry is all around us and is a fundamental property of nature. Syyymmetry and Introduction to Grou pyp Theory

The term symmetry is derived from the Greek word “symmetria” which means “measured together”. An object is symmetric if one part (e.g. one side) of it is the same* as all of the other parts. You know intuitively if somethinggy is symmetric but we re quire a precise method to describe how an object or molecule is symmetric.

Group theory is a very powerful mathematical tool that allows us to rationalize and simppylify man ypy problems in Chemistr y. A grou p consists of a set of symmetry elements (and associated symmetry operations) that completely describe the symmetry of an object.

We will use some aspects of group theory to help us understand the bonding and spectroscopic features of molecules. Space transformations. Translation symmetry.

" Crystallography is largely based Group Theory (symmetry). " Symmetry operations transform space into itself. Simplest symmetry operator is unity operator(=does nothing). (=Lattice is invariant with respect to symmetry operations)

" Translation operator, TR, replaces radius vector of every point, r, by r’=r+R. Symmetry a state in which parts on opposite sides of a plane, line, or point display arrangements that are related to one another via a symmetry operation such as translation, rotation, reflection or inversion. Divided into two ppyarts by a plane

Mirror plane (symmetry element)

Have no symmetry remaining : Asymmetric units AtiAsymmetric unit

Recall that the unit cell of a crystal is the smallest 3-D geometric figure that can be stacked without rotation to form the lattice. The asymmetric unit is the smallest part of a crystal structure from which the complete structure can be built using space group symmetry. The asymmetric unit may consist of only a part of a molecule, or it can contain more than one molecule, if the molecules not related by symmetry.

Protein Crystal Contacts by Eric Martz , April 2001 . http://molvis.sdsc.edu/protexpl/xtlcon.htm Group Theory

mathematical method a molecule’s symmetry

information about its properties (i.e ., structure , spectra , polarity, chirality, etc… )

A-B-A is different from A-A-B. O C

However, important aspects of the symmetry of H2O and CF2Cl2 are the same. Symmetry and Point Groups Point groups have symmetry about a single point at the center of mass of the system. Symmetry elements are geometric entities about which a symmetry operation can be performed. In a point group, all symmetry elements must pass through the center of mass (the point). A symmetry operation is the action that produces an object identical to the initial object. The symmetry elements and related operations that we will find in molecules are:

Element Operation

Rotation axis, Cn n-fold rotation

Improper rotation axis, Sn n-fold improper rotation Plane of symmetry, σ Reflection

Center of syyymmetry, i Inversion

Identity, E

The Identity operation does nothing to the object – it is necessary for mathematical completeness, as we will see later. There are two naming systems commonly used in describing symmetry elements

1The1. The Schoenflies notation used extensively by spectroscopists 2. The Hermann-Mauguin or international notation preferred by cystallographers Symmetry Operations/Elements

A molecule or object is said to possess a particular operation if that operation when applied leaves the molecule unchanged.

Each operation is performed relative to a point, line, or plane - called a symmetry element.

There are 5 kinds of operations

1. Iden tity 2. n-Fold Rotations 3. Reflection 4. Inversion 5. Improper n-Fold Rotation The 5 Symmetry Elements

Identity (C1 α E or 1) Rotation by 360° around anyyy randomly chosen axes throu gh the molecule returns it to original position. Any direction in any object is a C1 axis; rotation by 360 ° restores original orientation. Schoenflies : C is the syygmbol given to a rotational axis 1 indicates rotation by 360° Hermann-Mauguin : 1 for 1-fold rotation Operation: act of rotating molecule through 360° Element: axis of symmetry (i .e . the rotation axis) . Symmet r y Elem ents

Rotation turns all the points in the asymmetric unit around one point, the center of rottitation. A rot ttiation d oes not ch ange the handedness of figures in the plane. The center of rotation is the only invariant point (point that maps onto itself).

Good introductory symmetry websites http://mathforum.org/sum95/suzanne/symsusan.html http://www.ucs.mun.ca/~mathed/Geometry/Transformations/symmetry.html Rotation axes (Cn or n) Rotations through angles other than 360°. Operation: act of rotation Element: rotation axis Symbol for a symmetry element for which the operation is a rotation of 360°/n

C2 = 180° , C3 = 120° , C4 = 90° , C5 = 72° , C6 = 60° rotation, etc. Schoenflies : Cn Hermann–Mauguin : n. The molecule has an n-fold axis of symmetry. An objjypect may possess several rotation axes and in such a case the one ( or more) with the greatest value of n is called the principal axes of rotation. n-fold rotation - a rotation of 360°/n about the Cn axis (n = 1 to ∞)

O(1) 180° O(1)

H(2) H(3) H(3) H(2)

In water there is a C2 axis so we can perform a 2-fold (180°) rotation to get the identical arrangement of atoms.

H(3) H(4) H(2)

120° 120°

N(1) N(1) N(1)

H(4) H(2) H(4) H(3) H(2) H(3)

In ammonia there is a C3 axis so we can perform 3-fold (120°) rotations to get identical arrangement of atoms. The Principal axis in an objjgect is the highest order rotation axis. It is usually easy to identify the principle axis and this is typically assigned to the z-axis if we are using Cartesian coordinates.

Ethane, C2H6 Benzene, C6H6

The principal axis is the three-fold The principal axis is the six-fold axis axis containing the C-C bond. througggh the center of the ring.

The principal axis in a tetrahedron is a three-fold axis going through one vertex and the center of the object. Symmet r y Elem ents

Reflection

flips all points in the asymmetric unit over a line, which is called the mirror , and thereby changes the handedness of any figures in the asymmetric unit. The poi nt s al ong th e mi rror li ne are all invariant points (points that map onto themselves) under a reflection. Mirror p lanes ( ⌠ or m) Mirror reflection through a plane. Operation: act of reflection Element: mirror plane Schoenflies notation: Horizontal mirror plane ( ⌠h) : plane perpendicular to the principal rotation axis Vertical mirror plane ( ⌠v) : plane includes principal rotation axis Diagonal mirror pp(lane ( ⌠d))p : plane includes princi pal rotation axis and bisects the angle between a pair of 2-fold rotation axes that are normal (perpendicular) to principal rotation axis. ShSchoen flies : ⌠h, ⌠v, ⌠d Hermann–Mauguin : m Reflection across a plane of symmetry, σ (mirror plane)

O(1) σv O(1)

H(2) H(3) H(3) H(2)

These mirror planes are called “vertical” mirror

planes, σv, because they contain the principal axis.

O(1) σv O(1) The reflection illustrated in the top diagram is through a mirror plane H(2) H(3) H(2) H(3) perpendicular to the plane of the water molecule. The plane shown on the bottom is in the same plane as the Handedness is changed by reflection! water molecule. Notes about reflection operations: - A reflection operation exchanges one half of the object with the reflection of the other half.

- Reflection ppylanes may be vertical, horizontal or dihedral (more on σd later). - Two successive reflections are equivalent to the identity operation (nothing is moved). σ h A “horizontal” mirror plane, σh,is, is perpendicular to the principal axis. This must be the xy-plane if the z- axis is the ppprincipal axis. σ σ In benzene, the σh is in the plane d d of the molecule – it “reflects” each atom onto itself.

σh

Vertical and dihedral mirror planes of geometric shapes. σv σv Symmet r y Elem ents

Inversion every point on one side of a center of symmetry has a siilimilar poiitnt at an equal distance on the opposite side of the center of symmetry. CtCentres of fi inversi on ( cent re of symmet ry, i or 1 ) Operation: inversion through this point Element: point Straight line drawn through centre of inversion from any point of the molecule will meet an equi val ent poi nt at an equal di st ance b eyond th e cent er. Schoenflies : i Hermann–Mauguin : 1 Molecules possessing a center of inversion are centrosymmetric Inversion and centers of symmetry, i (inversion centers) In this operation, every part of the object is reflected through the inversion center, which must be at the center of mass of the object.

1 F Cl 22F Cl 1

2 Br i Br 1 1 221 12Br Br

1 Cl F22Cl F1

i [x, y, z] [-x, -y, -z]

We will no t cons ider the ma tr ix approac h to eac h o f the symme try opera tions in this course but it is particularly helpful for understanding what the inversion operation does. The inversion operation takes a point or object at [x, y, z] to [-x, -y, -z]. Aftii(iAxes of rotary inversion (improper rot ttiSation Sn or n ) Involves a combination or elements and operations Schoenflies : Sn Element: axis of rotatory reflection The operation is a combination of rotation by 360°/n (Cn) followed by reflection in a plane normal ( ⌠h) to the Sn axis.

Hermann-Mauguin : n Element: axis of rotatory inversion. The operation is a combination of rotation by 360 °/n (n) followed by inversion through a point (1). m n-fold improper rotation, Sn (associated with an improper rotation axis or a rotation-reflection axis) This operation involves a rotation of 360°/n followed by a reflection perpendicular to the axis. It is a single operation and is labeled in the same manner as “proper” rotations. F1 F2 H1 F F4 1 1 S4 H4 C H2

H3 F F 2 F 3 1 2 S4 F3 F4 F1 H2 90° σh H C H 1 1 3 C2

F3 H4 2 F4 S4

H3

H2 C H4

H1

Note that: S1 = σ, S2 = i, and sometimes S2n = Cn (e.g. in box) this makes more sense when you examine the matrices that describe the operations. Symmetry Elements in Arrays

Translational symmetry elements

Screw axis

: general symbol for a screw axis is Nn, where N is the order (2, 3, 4 or 6) of the axis

(()a) A twofold screw axis, 21

(b) A fourfold screw axis, 41 : SitittScrew axes are very common in protein structures

Glide plane (()c)

: combination of a mirror plane and a translation operation parallel to it Symmet r y Elem ents

Glide reflections reflects the asymmetric unit across a mirror line and then translates parallel to the mirror. A glide reflection changes the handedness of figures in the asymmetric unit . There are no invariant points (points that map onto themselves) under a glide reflection. Symmetry Elements

Screw axes rotation about the axis of symmetry by 360°/n, followed by a translation parallel to the axis b y r/ n of th e unit cell l ength in that direction.

Diagram from: http://www-structure.llnl.gov/xray/101index.html Symmet r y Elem ents

Translation moves all the points in the asymmetric unit the same distance in the same direction. This has no effect on the handedness of figures in the plane. There are no invariant points (points that map onto themselves) under a translation. A lesson in syyymmetry from M. C. Escher To be a group several conditions must be met :

1. The group multiplication table must be closed.

CidHConsider H2OhihhECO which has E, C2 and2d 2 sv's. Cmσm ≡ σm' i.e., 2 VV CCmm≡ El of course 22 etc…

The group multiplication table obtained is therefore:

E C2 σv σ'v

E E C2 σv σ'v

C2 C2 E σ'v σv

σv σv σ'v E C2

σ'v σ'v σv C2 E To be a group several conditions must be met :

2. Must have an identity ( El ).

3. All elements must have an inverse.

i.e., for a given operation ( lA ) there must exist an operation ( Bl ) such that llABE= l Point group (point symmetry)

: Allllhtidb32difftAll molecules characterized by 32 different combi bitifnations of symmetry elements

Space group ( point & translational symmetry)

: There are 230 possibl e arrangement s of symmet ry el ement s i n th e solid st at e. Any crystal must belong to one (and only one) space group.

There are two naminggy systems commonl y used in describin gyg symmetr y elements

1. The Schoenflies notation used extensively by spectroscopists 2. The Hermann-Mauguin or international notation preferred by cystallographers Typical space group symbols are: P 1 , C2/m, Ibca, R 3 , Fm3m, P212121.

P means priiiimitive,

A, B , or C means centred on the face perpendicular to the a, b or c axis,

F means centred on all the faces,

I means body centred -- centred in the middle of the cell --

R means rhombohedral, d) three-fold axis Reflection a) Mirror plane

Inversion n-Fold Rotations e) Center of b)) two-fold axis symmetry

c) two-fold axis + mirror planes

f) four-fold inversion axis The symmetry classification of molecules

The groups C1, Ci and Cs C1 : no element other than the identity Ci : identity and inversion alone Cn Cs : identity and a mirror plane alone

The groups Cn, Cnv and Cnh θ C4 Cn : n-fold rotation axis θ = n /2π Cnv : identity , Cn axis plus n vertical mirror planes ⌠v Cnh : identity and an n-fold rotation principal axis plus a horizontal mirror pllane ⌠h The groups Dn, Dnh and Dnd Dn : n-fold principal axis and n two-fold axes perpendicular to Cn Dnh : molecule also possesses a horizontal mirror plane Dnd : in addition to the elements of Dn possesses n dihedral mirror planes ⌠d

Cn

C2

C2 D4 C2 C2 The cubic groups Td and Oh : groups of the regular tetrahedron (e.g. CH4) and regular octahedron (e.g. SF6), respectively. T or O : object possesses the rotational symmetry of the tetrahedron or the octahedron, but none of their planes of reflection Th : based on T but also contains a center of inversion

The full rotation group R3 : consists of an infinite number of rotation axes with all possible values of n. A sphere and an atom belong to R3, but no molecule does. The groups Sn Sn : Molecules not already classified possessing one Sn axis S2 α Ci 32 Point Groups

Triclinic 1* -1

Monoclinic 2* m* 2/m

Orthorhombic 222 mm2* mmm

Tetragonal 4* -4 4/m 422 4mm* -42m 4/mmm

Trigonal 3* -3 32 3m* -3m

HlHexagonal 6* -6 6/m 622 6mm* -62m 6/mmm

Cubic 23 m-3 432 -43m m-3m

The 11 centrosymmetric point groups are shown in red, the 11 enantiomorphic point groups are shown in green, and the 10 ppgppolar point groups are shown with an asterisk *. ((,Note that in the above table, inversion axes are written with a minus sign in front of the axis symbol.) 230 space groups 1-2 : Triclinic, classes 1 and -1 3-15 : Monoclinic, classes 2, m and 2/m 16-24 : Orthorhombic, class 222 25-46 : Orthorhombic, class mm2 47-74 : Orthorhombic, class mmm 75-82 : Tetragonal, classes 4 and -4 83-88 : Tetragonal, class 4/m 89-98 : Tetragonal, class 422 99-110 : Tetragonal, class 4mm 111-122 : T etragonal , cl ass -42m 123-142 : Tetragonal, class 4/mmm 143-148 : Trigonal, classes 3 and -3 149-155 : T ri gonal , cl ass 32 156-161 : Trigonal, class 3m 162-167 : Trigonal, class -3m 168-176 : Hexagonal, classes 6 , -6 and 6/m 177-186 : Hexagonal, classes 622 and 6mm 187-194 : Hexagonal, classes -6m2 and 6/mmm 195-206 : Cubic, classes 23 and m -3 206-230 : Cubic, classes 432, -43m and m-3m Triclinic system

Center of symmetry MliitMonoclinic system Trigonal system

32 P oi nt G roups

7 Crystal Systems

orthorhombic hexagonal monoclinic trigonal cubi c tetragonal triclinic

Crystal System External Minimum Symmetry Unit Cell Properties Triclinic None a, b, c, al, be, ga, Monoclinic One 2-fold axis, || to b (b unique) a, b, c, 90, be, 90 Orthorhombic Three perpendicular 2 -folds a, b, c, 90, 90, 90 Tetragonal One 4-fold axis, parallel c a, a, c, 90, 90, 90 Trigonal One 3-fold axis a, a, c, 90, 90, 120 Hexagonal One 6-fold axis a, a, c, 90, 90, 120 Cubic Four 3-folds along space diagonal a, a, ,a, 90, 90, 90 The combination of all available symmetry operations (point groups plus glides and screws) within the seven crystal systems equals 230 combinations, called the 230 Space Groups. The International Tables list those by symbol and number, together with symmet ry operat ors, ori gi ns, reflection conditions, and space group projection diagrams.

Inversion symmetry elements are not allowed when dealinggp with protein crystals (all amino acids present in proteins have the L stereochemical configuration; the inverse, the D configuration, can’t be found in Page 151, Vol. I. from proteins.) Therefore, the number of International Tables for X-Ray space groups is reduced from 230 for Cryygpy,stallography, 1965 edition small molecules to 65 for proteins. Identification of the Spppace Group is called indexing the crystal. The International Tables for X-ray Crystallography tell us a huge amount of information about any given space group. For instance, If we look up space group P2, we fin d it has a 2-fldfold rotation axis and the following symmetry equivalent positions:

X , Y , Z -X , Y , -Z and an asymmetric unit defined by:

0 ≤ x ≤ 1 0 ≤ y ≤ 1 0 ≤ z ≤ 1/2

An interactive tutorial on Space Groups can be found on-line in Bernhard Rupp’s Crystallography 101 Course: http://www-structure. llnl.gov/Xray/tutorial/spcgrps .htm